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The universal property of commutative algebras' internal crossed modules

Yıl 2023, , 77 - 85, 31.08.2023
https://doi.org/10.54187/jnrs.1281267

Öz

In this work, we identify subobjects and ideals in the category of internal crossed modules, which provide a deeper understanding of the structure of these objects. Moreover, we provide several propositions through examples, which illustrate the properties and relationships between ideals and subobjects in the category of internal crossed modules. The examples and propositions provided in this work can serve as a foundation for further research in this area and may lead to new insights and discoveries in the study of these complex algebraic structures. Overall, in conclusion, we give a brief overview of the contributions and future research directions of the work presented, highlighting the significance of internal crossed modules in algebraic topology and category theory as well as making suggestions for possible areas of additional research and application.

Kaynakça

  • J. H. C. Whitehead, Combinatorial homotopy II, Bulletin of American Mathematical Society 55 (5) (1949) 453-496.
  • S. Lichtenbaum, M. Schlessinger, The cotangent complex of a morphism, Transactions of the American Mathematical Society 128 (1) (1967) 41-70.
  • R. Brown, J. Huebschmann, Identities among relations. Low dimensional topology, London Mathematical Society Lecture Notes 46 (1982) 153-202.
  • R. Brown, C. B. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, In Indagationes Mathematicae (Proceedings) 79 (4) (1976) 296-302.
  • T. Porter, Some categorical results in the theory of crossed modules in commutative algebras, Journal of Algebra 109 (2) (1987) 415-429.
  • G. J. Ellis, Higher dimensional crossed modules of algebras, Journal of Pure and Applied Algebra 52 (3) (1988) 277-282.
  • E. S. Yılmaz, K. Yılmaz, On crossed squares of commutative algebras, Mathematical Sciences and Applications E-Notes 8 (2) (2020) 32-41.
  • Z. Arvasi, Crossed squares and 2-crossed modules of commutative algebras, Theory and Applications of Categories 3 (7) (1997) 160-181.
  • K. Yılmaz, 2-Crossed modules over same base, Sakarya University Journal of Science 21 (6) (2017) 1210-1220.
  • G. Ellis, Crossed squares and combinatorial homotopy, Mathematische Zeitschrift 214 (1) (1993) 93-110.
  • K. Yılmaz, A fibration application for crossed squares, Cumhuriyet Science Journal 39 (1) (2018) 1-6.
  • P. Carrasco, J. M. Moreno, Categorical G-crossed modules and 2-fold extensions, Journal of Pure and Applied Algebra 163 (3) (2001) 235-257.
  • G. Janelidze, Internal crossed modules, Georgian Mathematical Journal 10 (1) (2003) 99-114.
  • T. Porter, Extensions, crossed modules and internal categories in categories of groups with operations, Proceedings of the Edinburgh Mathematical Society 30 (3) (1987) 373-381.
  • M. Forrester-Barker, Representations of crossed modules and cat$\pi$-groups, Doctoral Dissertation University of Wales (2003) Bangor.
  • E. S. Yılmaz, U. E. Arslan, Internal cat-1 and XMod, Journal of New Theory (38) (2022) 79-87.
  • U. E. Arslan, E. S. Yılmaz, On internal categories in higher dimensional groups, Journal of Universal Mathematics 5 (2) (2022) 129-138.
  • N. M. Shammu, Algebraic and categorical structure of categories of crossed modules of algebras, Doctoral Dissertation University of North Carolina Wilmington (1992) Bangor.
  • D. Conduché, Modules croisés généralisés de longueur 2, Journal of Pure and Applied Algebra 34 (2) (1984) 155-178.
  • A. R. Grandjeán, M. J. Vale, 2-Modulos cruzados en la cohomologia de André-Quillen, Memorias de la Real Academia de Ciencias 22 (1986) 1-28.
  • A. Odabaş, E. Soylu, Relationships between category theory and functional programming with an application, Turkish Journal of Mathematics 43 (3) (2019) 1566-1577.
  • M. Alp, C. D. Wensley, Automorphisms and homotopies of groupoids and crossed modules, Applied Categorical Structures 18 (5) (2010) 473-504.
  • A. Mutlu, Peiffer Commutators by using GAP package, Mathematical and Computational Applications 3 (1) (1998) 59-65.
  • B. Ekici, Formal categorical reasoning, Turkish Journal of Mathematics 46 (4) (2022) 1538-1552.
Yıl 2023, , 77 - 85, 31.08.2023
https://doi.org/10.54187/jnrs.1281267

Öz

Kaynakça

  • J. H. C. Whitehead, Combinatorial homotopy II, Bulletin of American Mathematical Society 55 (5) (1949) 453-496.
  • S. Lichtenbaum, M. Schlessinger, The cotangent complex of a morphism, Transactions of the American Mathematical Society 128 (1) (1967) 41-70.
  • R. Brown, J. Huebschmann, Identities among relations. Low dimensional topology, London Mathematical Society Lecture Notes 46 (1982) 153-202.
  • R. Brown, C. B. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, In Indagationes Mathematicae (Proceedings) 79 (4) (1976) 296-302.
  • T. Porter, Some categorical results in the theory of crossed modules in commutative algebras, Journal of Algebra 109 (2) (1987) 415-429.
  • G. J. Ellis, Higher dimensional crossed modules of algebras, Journal of Pure and Applied Algebra 52 (3) (1988) 277-282.
  • E. S. Yılmaz, K. Yılmaz, On crossed squares of commutative algebras, Mathematical Sciences and Applications E-Notes 8 (2) (2020) 32-41.
  • Z. Arvasi, Crossed squares and 2-crossed modules of commutative algebras, Theory and Applications of Categories 3 (7) (1997) 160-181.
  • K. Yılmaz, 2-Crossed modules over same base, Sakarya University Journal of Science 21 (6) (2017) 1210-1220.
  • G. Ellis, Crossed squares and combinatorial homotopy, Mathematische Zeitschrift 214 (1) (1993) 93-110.
  • K. Yılmaz, A fibration application for crossed squares, Cumhuriyet Science Journal 39 (1) (2018) 1-6.
  • P. Carrasco, J. M. Moreno, Categorical G-crossed modules and 2-fold extensions, Journal of Pure and Applied Algebra 163 (3) (2001) 235-257.
  • G. Janelidze, Internal crossed modules, Georgian Mathematical Journal 10 (1) (2003) 99-114.
  • T. Porter, Extensions, crossed modules and internal categories in categories of groups with operations, Proceedings of the Edinburgh Mathematical Society 30 (3) (1987) 373-381.
  • M. Forrester-Barker, Representations of crossed modules and cat$\pi$-groups, Doctoral Dissertation University of Wales (2003) Bangor.
  • E. S. Yılmaz, U. E. Arslan, Internal cat-1 and XMod, Journal of New Theory (38) (2022) 79-87.
  • U. E. Arslan, E. S. Yılmaz, On internal categories in higher dimensional groups, Journal of Universal Mathematics 5 (2) (2022) 129-138.
  • N. M. Shammu, Algebraic and categorical structure of categories of crossed modules of algebras, Doctoral Dissertation University of North Carolina Wilmington (1992) Bangor.
  • D. Conduché, Modules croisés généralisés de longueur 2, Journal of Pure and Applied Algebra 34 (2) (1984) 155-178.
  • A. R. Grandjeán, M. J. Vale, 2-Modulos cruzados en la cohomologia de André-Quillen, Memorias de la Real Academia de Ciencias 22 (1986) 1-28.
  • A. Odabaş, E. Soylu, Relationships between category theory and functional programming with an application, Turkish Journal of Mathematics 43 (3) (2019) 1566-1577.
  • M. Alp, C. D. Wensley, Automorphisms and homotopies of groupoids and crossed modules, Applied Categorical Structures 18 (5) (2010) 473-504.
  • A. Mutlu, Peiffer Commutators by using GAP package, Mathematical and Computational Applications 3 (1) (1998) 59-65.
  • B. Ekici, Formal categorical reasoning, Turkish Journal of Mathematics 46 (4) (2022) 1538-1552.
Toplam 24 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Elis Soylu Yılmaz 0000-0002-0869-310X

Yayımlanma Tarihi 31 Ağustos 2023
Yayımlandığı Sayı Yıl 2023

Kaynak Göster

APA Soylu Yılmaz, E. (2023). The universal property of commutative algebras’ internal crossed modules. Journal of New Results in Science, 12(2), 77-85. https://doi.org/10.54187/jnrs.1281267
AMA Soylu Yılmaz E. The universal property of commutative algebras’ internal crossed modules. JNRS. Ağustos 2023;12(2):77-85. doi:10.54187/jnrs.1281267
Chicago Soylu Yılmaz, Elis. “The Universal Property of Commutative algebras’ Internal Crossed Modules”. Journal of New Results in Science 12, sy. 2 (Ağustos 2023): 77-85. https://doi.org/10.54187/jnrs.1281267.
EndNote Soylu Yılmaz E (01 Ağustos 2023) The universal property of commutative algebras’ internal crossed modules. Journal of New Results in Science 12 2 77–85.
IEEE E. Soylu Yılmaz, “The universal property of commutative algebras’ internal crossed modules”, JNRS, c. 12, sy. 2, ss. 77–85, 2023, doi: 10.54187/jnrs.1281267.
ISNAD Soylu Yılmaz, Elis. “The Universal Property of Commutative algebras’ Internal Crossed Modules”. Journal of New Results in Science 12/2 (Ağustos 2023), 77-85. https://doi.org/10.54187/jnrs.1281267.
JAMA Soylu Yılmaz E. The universal property of commutative algebras’ internal crossed modules. JNRS. 2023;12:77–85.
MLA Soylu Yılmaz, Elis. “The Universal Property of Commutative algebras’ Internal Crossed Modules”. Journal of New Results in Science, c. 12, sy. 2, 2023, ss. 77-85, doi:10.54187/jnrs.1281267.
Vancouver Soylu Yılmaz E. The universal property of commutative algebras’ internal crossed modules. JNRS. 2023;12(2):77-85.


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